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As you might discern from previous posts, I have been teaching myself:

- Calculus on manifolds
- Differential forms
- Lie Algebra, Group
- Push forward, pull back.

I am an engineer approaching this late in life and with a deficient background in math. It is all coming together and I almost see the picture. But it remains discrete topics - not connected with each other and without a direct connection to my deficienies.

This post concerns THE GLUE.

Could someone explain how all these ideas are integrated, in WORDS and with an acknowledgment to a person with deficient math (math learned in a country with a math education a mile wide and an inch deep).

Essentially, IN WORDS (not symbols)...

And YES, these are huge questions, but I am not interested in direct answers to each but in the continuity between the questions -- oh hell, I wish I knew exactly how to say this: my calculus was so bad I practically need someone to say: "In a Lagrangian there are generalized coordinates which become the manifold and there are vectors in the tangent plane (position) and can only be multiplied by co-vectors (velocities) and the purpose of all this is only to see the structure..." Or something like that...

Thus...

Why should I learn calculus on manifolds (ConM)?

What impact do they have in Lagrange's equation for a dynamical system?

What can ConM do that my “normal” calculus cannot?

Why do forms matter?

How do form really help do math on the tangent space?

What is the relationship between forms and vectors?

Why is it that all treatments on this also treat Lie Algebra and Groups?

Lie groups are just smooth groups, so where is the manifold.?

I know it is going to be difficult to do this – write an answer in words.

Thank you if you can. (And, yes, as evidenced by previous posts: I am looking for words... for only by WORDS can I see where I was miseducated.)